Welcome to block 2 linear algebra.
My name is Rosa Estela
and I'm going to present this video talking about matrices.
The content of this video
is a little introduction of matrices,
the definition, representation
and then matrix operations: addition,
subtraction and scalar multiplication.
The definition of transpose
and symmetric matrices.
The operation of matrix multiplication
properties and rules. The definition
of inverse matrices
and then how to solving system
of linear equations using matrices.
Why do we need to learn matrices?
The aim of a matrix is to think through the results expected
from a project in a logical order. It helps
in solving linear equations.
And matrices are very important
in mathematics because facilitates the representation,
transformation and analysis of data.
As a instruction
the term matrix was introduced by the 19th-century
by the English mathematician James Sylvester.
But is it what his friend,
the mathematician Arthur Cayley, who developed the algebraic
aspect of matrices in two papers in 1850s.
The role of matrices in linear
algebra and its application is so huge that it can be said
that is the soul of linear algebra. The applications of matrices
you could see in
other fields
of science and real life
such as Probability Theory and Statistics, Electronics,
Engineering, Computer Science, Cryptography,
Wireless Communication and
in all the fields.
First, the definition of matrix.
A matrix is a rectangular array of numbers,
called the entries of the matrix.
The horizontal and vertical lines of entries
in a matrix are called rows and columns,
respectively. An m by n matrix
is a rectangular grid of numbers
with m rows and n columns.
A column vector is an m by 1 matrix.
A row vector is a 1 by n matrix,
and a square matrix is a matrix
m by m for some natural m.
The representation of the matrix
m by n is like you can see
it's a numbers in columns and rows.
This matrix has m rows
and n columns.
Let's see the operations about matrices.
First of all, addition. Let A and B
be matrices of the same dimension m X n
and let c be any real number.
The sum A+B
is another m by n matrix
obtained by adding corresponding entries of A and B.
The difference A-B is another
m by n matrix obtained by subtracting
corresponding entries of A and B. And the scalar product,
is another m by n matrix
obtained by multiplying each entry of A by C.
After,
in the next video we will show you some examples.
Definition of transpose
and symmetric matrices.
The matrix formed by interchanging the rows
and columns of a matrix A, is called the transpose of A
and is denoted by A^T.
Now,
as an example,
calculate the transpose of the matrix A
with these numbers.
See that the transpose is, in A
the first row is 3 1
and now the first column in the transpose matrix is 3 1
and the same with the second row.
So let's define symmetric matrix.
You need to have a square matrix A with the property,
A equal to the transpose of A,
you say A is symmetric.
Let's see another important operation
matrix multiplication.
Let be A and B
two matrices with dimension
A m by n
and B n by k
See that the columns in A
is the same number that rows in B. This is important,
if not, we can't do the operation.
So if A is an m by n matrix
and B is an n by k matrix
then their product is another
m by k matrix where
c_ij is the inner product of the
ith row of A and the the jth column of B,
and we write the product as
C=AB
And as we said before,
this is the calculus of the matrix
multiplication C
given one row
and another column, we calculate the inner product.
We will practice matrix multiplication in the next video.
Let's see matrix
multiplication properties.
Let A,B and C be matrices,
then A(BC)=(AB)C is called associative property
and distributive property
is A(B+C)=AB+AC, (B+C)A=BA+CA.
for matrices which the products are defined.
And important,
the operation of matrix multiplication
is not commutative.
Even when both AB and BA are defined,
they aren't necessary equal.
Let's see an example.
Let be A this matrix and B the other one
that are two by two matrices
so we can calculate the product AB
and BA.
If you calculate like we have seen before,
we have AB this matrix
and BA this matrix, you can see that
the matrices are not the same because two
matrices are equal if they have the same numbers in the corresponding positions
and this is not our case.
So in general AB is different to BA.
So matrix multiplication is not commutative.
Let's talk about inverse matrices.
First of all, we need to define
the identity matrix.
The identity matrix is the
nxn matrix for which
each main diagonal entry is a 1,
and all other entries are 0.
We have here three examples
in dimension two, three and four.
The identity matrix holds this property,
if we multiply
a matrix by the identity,
the result is the same initial matrix.
So now we know whose is the identity matrix,
let's define the inverse matrices.
Let A be a square nxn matrix,
if there exists an nxn matrix A^-1
with this property,
then we say that this new matrix
is the inverse of A.
As an example, let be A this two by two matrix.
Then the inverse matrix
is this one.
Next I will show you how to calculate,
but you can see that the product
in both directions gives the identity matrix,
in this case, the 2x2 identity matrix.
So, how can we find
the inverse of an nxn matrix?
Let be A an nxn matrix.
First of all, we construct another matrix
n x 2n matrix with the entries of A
on the left of the new matrix
and the identity matrix I_n on the right.
Then we use the elementary
row operations on this new large matrix
to change the left side into the identity matrix.
The right side is trasformed automatically into the inverse matrix.
We will show you some examples
in the next video.
And why is this needed?
For example, by solving systems of linear equations.
Here we
have one system of linear equations
that we can write
using matrices like this,
because if we multiply this,
we have the initial system.
For example, we have a system like this
with these real numbers,
and A is called the coefficient matrix.
Then we can solve the system
because the solution X is the product
of the inverse of the given matrix A by B
and we solve the system.
And here you have
the references for more study of matrices.
You have three books,
and also I put here Wolfram Alpha,
CalcMe of wiris,
because you can calculate
quickly with matrices
and here you have information also of the Engimath European project
is a project who only work
with matrices and is an extensive project.
And then you have also the MOOC "El Lenguaje de la IngenierĂ­a"
where you can find more information
about this today topic, matrices.
So thank you, and enjoy practicing with this.